Linear Algebra true/false explanation.

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Homework Help Overview

The discussion revolves around a true/false question in linear algebra concerning the intersection of two-dimensional subspaces in R3. The original poster is exploring whether it is possible for the intersection of two such subspaces to be solely the zero vector.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the implications of the intersection of subspaces, referencing linear combinations and linear independence. Other participants question the geometric interpretation of the problem, specifically whether two planes can intersect only at the origin.

Discussion Status

The conversation is ongoing, with participants exploring the geometric properties of subspaces and clarifying definitions. Some guidance has been offered regarding the meaning of the intersection being equal to the zero vector, but no consensus has been reached on the original question.

Contextual Notes

The original poster references textbook material and definitions related to linear independence, indicating a reliance on course resources for understanding. There is an implicit assumption that the participants are familiar with the concepts of subspaces and their intersections.

A_lilah
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Linear Algebra true/false explanation. :)

Homework Statement



True or False:
Is it possible to find a pair of two-dimensional subspaces S and T of R3 such that S (upside down U) = {0} ?

Homework Equations





The Attempt at a Solution



My understanding: upside down U = intersection, and for S to intersect T, S must have some vectors that are linear combinations of the vectors of T (and vice versa). These combinations have to = 0... and that's all I've got. I've sifted through my notes and the textbook several times, and the only other thing I could come up with that may or may not be useful (I haven't connected the dots yet), is that:
Def: the vectors v1, v2,...,vn in a vector space V are said to be linearly independent if
c1v1+c2v2+...+cnvn = 0
which implies that all the scalars c1...cn must equal zero

(the answer in the back of the book says that the answer is false, which is why I was looking at linear independence.)
Thanks!
 
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Think about the geometry of your problem. S and T are both two-dimensional subspaces of R^3, which means that both (S and T) are planes that contain the origin. Is it possible to have two planes in R^3, both containing the origin, that intersect at no other points?
 


No. So does = {0} mean they only contain the origin then?
 


Yes, {x} means a set that contains only the single member x. Surely you knew that?
 

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